Re: Calculus (General) Formulas

Ricky wrote:

limit as x approaches 0 of tan(1/x) is undefined.

Same goes with arctan.

I don't think thats correct. Yes the limit of tan (1/x) as x approaches zero is undefined, but the limit of arctan (1/x) as x approaches zero IS pi/2. Think about it, if θ is an angle in a right triangle, and the side opposite θ is 1 and the side adjacent to θ is x, as x approaches zero, θ aproaches 90 from the left.

Re: Calculus (General) Formulas

This isn't rigourous, but more of a way to think about it. limit 1/x as x goes to 0 from the positive side is positive infinity. So what we really want to find is the limit of arctan(x) as x goes to infinity, which is pi/2.

But what about when we approach 1/x from the negative side? Then it's negative infinity, and thus, we want to find the limit of arctan(x) as x approaches negative infinity. That's -pi / 2.

Since the limit from the left is not the limit from the right, the limit does not exist.

For a clear way to see it, just use a graphing calculator. Or just do arctan(1/.000000001) and arctan(1/-000000001).

Last edited by Ricky (2006-04-10 07:25:59)

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

The Bernoulli numbers B[sub]n[/sub] are a sequence of special rational numbers. Their derivation is outside the scope of this thread, but an explanation may be offered elsewhere upon sufficient demand. The first few Bernoulli numbers are(note that for odd n other than 1, B[sub]n[/sub] = 0):

E[sub]n[/sub]: The Euler Numbers

The Euler numbers E[sub]n[/sub] are a sequence of special numbers. Their derivation is outside the scope of this thread, but an explanation may be offered elsewhere upon sufficient demand. The first few Euler numbers are(note that for all odd n, E[sub]n[/sub] = 0):